Mass matrix acts on the correlation between position variables

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Hamiltonian Monte Carlo

Assume we have an estimate \(\Sigma\) of the covariance matrix for \(q\). One way to improve sampling is to transform the \(q\) so their covariance matrix is close to identity. To do this we write the Cholesky decomposition:

\begin{equation*} \Sigma = L^T\,L \end{equation*}

and write \(q' = L^{-1}\,q\). If we use \(K(p) = p^{T}\,p\;/2\) we have independent momenta, almost independent positions and HMC should perform well with a small number of leapfrog steps. This transformation is equivalent to using \(\Sigma\) as the inverse mass matrix:

\begin{equation*} K(p) = p^{T}\,\Sigma\,p\; / 2 \end{equation*}